sim.data.ts.single {gma}

Generate single-level simulation data

Description

This function generates a single-level dataset with given parameters.

Usage

sim.data.ts.single(n, Z, A, B, C, Sigma, W, Delta = NULL, p = NULL, nburn = 100)

Arguments

n

an integer indicating the length of the time series.

Z

a vector of treatment/exposure assignment at each time point.

A

a numeric value of model coefficient.

B

a numeric value of model coefficient.

C

a numeric value of model coefficient.

Sigma

a 2 by 2 matrix, is the covariance matrix of the two Gaussian white noise processes.

W

a 2p by 2 matrix, is the transition matrix.

Delta

a 2 by 2 matrix, is the covariance matrix of the initial condition. Default is NULL, will be the same as Sigma.

p

a numeric value indicating the order of the vector autoregressive (VAR) model. Default is NULL, will be calculated based on W.

nburn

a integer indicating the number of burning sample. Default is 100.

Details

The single level GMA model is

M_{t}=Z_{t}A+E_{1t},

R_{t}=Z_{t}C+M_{t}B+E_{2t},

and for stochastic processes (E_{1t},E_{2t}),

E_{1t}=\sum_{j=1}^{p}\omega_{11_{j}}E_{1,t-j}+\sum_{j=1}^{p}\omega_{21_{j}}E_{2,t-j}+\epsilon_{1t},

E_{2t}=\sum_{j=1}^{p}\omega_{12_{j}}E_{1,t-j}+\sum_{j=1}^{p}\omega_{22_{j}}E_{2,t-j}+\epsilon_{2t}.

Sigma is the covariance matrix of the Gaussian white noise (\epsilon_{1t},\epsilon_{2t}), and Delta is the covariance matrix of (\epsilon_{10},\epsilon_{20}). W is the transition matrix with element \omega's.

Value

The function returns a list with two data frames. One is the data with variables Z, M and R; one is the data frame of (E_{1t},E_{2t}).

Author(s)

Yi Zhao, Brown University, [e-mail]; Xi Luo, Brown University, [e-mail]

References

Zhao, Y., & Luo, X. (2017). Granger Mediation Analysis of Multiple Time Series with an Application to fMRI. arXiv preprint arXiv:1709.05328.

Examples

###################################################
# Generate a single-level dataset

# covariance matrix of errors
delta<-0.5
Sigma<-matrix(c(1,2*delta,2*delta,4),2,2)

# model coefficients
A0<-0.5
B0<--1
C0<-0.5

# number of time points
n<-500

# generate a treatment assignment vector
set.seed(1000)
Z<-matrix(rbinom(n,size=1,prob=0.5),n,1)

# VAR(1) model
p<-1

# Delta and W matrices
Delta<-matrix(c(2,delta*sqrt(2*8),delta*sqrt(2*8),8),2,2)
W<-matrix(c(-0.809,0.154,-0.618,-0.5),2,2)

# number of burning samples
nburn<-1000

set.seed(1000)
data.single<-sim.data.ts.single(n,Z,A0,B0,C0,Sigma,W,Delta,p=p,nburn=nburn)
###################################################